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Topics In Primitive Roots N. A. Carella Abstract: This monograph considers a few topics in the theory of primitive roots modulo a prime p≥ 2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g *(p) modulo p, a large prime, are determined. One of the estimate here seems to sharpen the Burgess estimate g(p)≪p 1/4+ϵ for arbitrarily small number ϵ> 0, to the smaller estimate g(p)≪p 5/loglogp uniformly for all large primes p⩾ 2. The expected order of magnitude is g(p)≪(logp) c, c> 1 constant. The corre- sponding estimates for least prime primitive roots g*(p) are slightly higher. The last topics deal with Artin conjecture and its generalization to the set of integers. Some effective lower bounds such as {p⩽x : ord(g)=p-1}≫x(logx) -1 for the number of primes p⩽x with a fixed primitive root g≠±1,b 2 for all large number x⩾ 1 will be provided. The current results in the literature have the lower bound {p⩽x : ord(g)=p-1}≫x(logx) -2, and have restrictions on the minimal number of fixed integers to three or more. Mathematics Subject Classifications: Primary 11A07, Secondary 11Y16, 11M26. Keywords: Prime number; Primitive root; Least primitive root; Prime primitive root; Artin primitive root conjecture; Cyclic group. Copyright 2015 ii Table Of Contents 1. Introduction 1.1. Least Primitive Roots 1.2. Least Prime Primitive Roots 1.3. Subsets of Primes with a Fixed Primitive Roots 1.4. Subsets of Composites with a Fixed Primitive Roots 1.5. Multiplicative Subsets of Composites with Fixed Primitive Roots 2. Characters And Exponential Sums 2.1 Simple Characters Sums 2.2 Estimates of Exponential Sums 2.3 Some Exact Evaluations 3. Representations of the Characteristic Function 3.1 Divisors Dependent Characteristic Functions 3.2 Divisors Free Characteristic Functions 3.3 Moduli Free Characteristic Function for Finite Rings 3.4 Characteristic Function for Finite Rings 3.5 Quasi Characteristic Functions 3.6 Basic Statistics of the Finite Rings 3.7 Summatory Functions 4. Prime Divisors Counting Functions 4.1 Average Orders 4.2 Distribution of Prime Divisors 4.3 Problems And Exercises 5. Euler Totient Function 5.1 Estimates and Average Orders of the Euler Totient Function 5.2 Average Gap 5.3 Density of the Euler Totient Function 5.4 Range of Values of the Totient Function 5.5 Average Order of the Number of Primitive Roots in the Multiplicative Group 5.6 Multiplicity of the Totient Function 5.7 Squarefree Values of the Totient Function 5.8 Problems And Exercises Copyright 2015 iii 6. Finite Sums And Products Of Totients 6.1 Finite Products over the Primes 6.2 The Squarefree Totients 6.3 Moments of the Density Function 6.4. Evaluation Of A Finite Sum 6.5 Problems And Exercises 7. Carmichael Function 7.1 Estimates and Average Orders of the Carmichel Function 7.2 Range of Values of the Carmichael Function 7.3 Average Order of the Number of Primitive Roots in Subgroups of the Multiplicative Group 7.4 Average Gap 7.5 Squarefree Values of the Carmichel Function 8. Basic L-Functions Estimates 8.1 An L-Function for the Least Primitive Roots 8.2 An L-Function for the Least Prime Primitive Roots 8.3 An L-Function for Fixed Primitive Roots 9. Least Primitive Roots 9.1 The Proof 9.2 Problems And Exercises 10. Least Prime Primitive Roots 10.1. The Proof 10.2. Problems And Exercises 11. Artin Conjecture 11.1 Conditional result 11.2 Average Results 11.3. Current Results 12. Subsets of Primes with Fixed Primitive Roots 12.1 A Proof of the Result 12.2 The Densities Of Fixed Primitive Roots In Arithmetic Progressions. 12.3 The Densities of A Few Fixed Primitive Roots. Copyright 2015 iv 13. Subsets of Integers with Fixed Primitive Roots 13.1 First Approach to the Subsets of Integers Fixed Primitive Roots 13.2 Second Approach to the Subsets of Integers With Fixed Primitive Roots 14. Other Densities Problems 15. Harmonic Sums 15.1. Harmonic Sums Over The Integers 15.2 Harmonic Sums Over Large Subsets of Integers 15.3. Harmonic Sums Over Squarefree Integers 15.4 Harmonic Sums For Fixed Primitive Roots 15.5 Harmonic Sums For Fixed Quadratic Residues 15.6. Fractional Finite Sums Over The Integers 15.7. Problems And Exercises 16. Prime Numbers Theorems 16.1 Asymptotic Formulas 16.2 Densities Of Primes In irrational Sequences 17. Prime Harmonic Sums Over Primes 17.1 Prime Harmonic Sums Over The Primes 17.2 Prime Harmonic Sums Over Large Subsets 17.3 Prime Harmonic Sums Over Subsets Of Arithmetic Progressions 17.4 Prime Harmonic Sums Over Primes With Fixed Quadratic Nonresidues 17.5 Prime Harmonic Sums Over Primes With Fixed Cubic Nonresidues 17.6 Prime Harmonic Sums Over Primes With Fixed Quartic Nonresidues 17.7 Prime Harmonic Sums Over Primes With Fixed Primitive Roots 17.8 Prime Harmonic Sums Over Squarefree Totients 17.9. Fractional Finite Sums Over The Primes 17.10. Problems And Exercises 18. Connecting Constants 18.1 Definitions 18.2 Problems And Exercises 19. Prime Harmonic Products 19.1 Prime Harmonic Products Over The Primes 19.2 Prime Harmonic Products Over Large Subsets Of Primes 19.3 Prime Harmonic Products For Fixed Primitive Roots Copyright 2015 v 19.4 Prime Harmonic Products Over Arithmetic Progressions 19.5 Prime Harmonic Products For Fixed Primitive Roots In Arithmetic Progressions 19.6 Prime Harmonic Products For Fixed Quadratic Residues 19.7 Problems And Exercises 20. Primes Decompositions Formulae 20.1. Wirsing Formula 20.2. Rieger Formula 20.3. Delange Formula 20.4. Halasz Formula 21. Characteristic Functions In Finite Rings 21.1 The Characteristic Function of Quadratic Nonresidues in the Finite Rings 21.2 The Characteristic Function of Cubic Nonresidues in the Finite Rings 21.3 The Characteristic Function of Quartic Nonresidues in the Finite Rings 21.4 The Characteristic Function of Primitive Roots in the Finite Rings 21.5. Problems And Exercises 22. Primitive Roots In Finite Rings 22.1 Primitive Roots Test and Primality Tests 22.2 Abel, and Wieferich Primes 22.3 Correction Factor 22.4. Problems And Exercises 23. Generalized Artin Conjecture 23.1 Subset of Integers With Fixed Primitive Root 2 23.2. Subset of Integers With Fixed Primitive Root u 23.3. Problems And Exercises 24. Asymptotic Formulae For Quadratic, Cubic, And Quartic Nonresidues 24.1 Subset of Integers With Fixed Quadratic Nonresidues 24.2. Subset of Integers With Fixed Cubic Nonresidues 24.3. Subset of Integers With Fixed Quartic Nonresidues 24.4. Problems And Exercises 25. The L-Functions Of Fixed Primitive Roots 25.1 Problems And Exercises Copyright 2015 vi 26. Class Numbers And Primitive Roots 26.1 Descriptions and Results 26.1 Problems And Exercises 27. Appendix 28. References Copyright 2015 vii Chapter 1 Introduction A few topics in the theory of primitive roots modulo primes p≥ 2, and primitive roots modulo integers n≥ 2, are studied in this monograph. The topics investigated are listed below. 1.1. Least Primitive Roots Chapter 9 deals with estimates of the least primitive roots g(p) modulo p, a large prime. One of the estimate here seems to sharpen the Burgess estimate g(p)≪p 1/4+ϵ (1.1) for arbitrarily small number ϵ> 0, to the smaller estimate g(p)≪p 5/loglogp (1.2) uniformly for all large primes p⩾ 2. The expected order of magnitude is g(p)≪(logp) c, c> 1 constant. The result is stated in Theorem 9.1. 1.2. Least Prime Primitive Roots Chapter 10 provides the details for the analysis of some estimates for the least prime primitive root g*(p) in the cyclic group ℤ/(p-1)ℤ,p≥ 2 prime. The current literature has several estimates of the least prime primitive root g *(p) modulo a prime p⩾ 2 such as g*(p)≪p c,c> 2.8. (1.3) The actual constant c> 2.8 depends on various conditions such as the factorization of p- 1, et cetera. These results are based on sieve methods and the least primes in arithmetic progressions, see [MA98], [MB97], [HA13]. Moreover, there are a few other conditional estimates such as g(p)⩽g *(p)≪(logp) 6, see [SP92], and the conjectured upper bound g*(p)≪(logp)(loglogp) 2, see [BH97]. On the other direction, there is the Turan lower bound g*(p)⩾g(p) =Ω(logp loglogp), refer to [BE68], [RN96, p. 24], and [MT91] for discussions. The result stated in Theo- rem 10.1 improves the current estimate to the smaller estimate Page 1 Topics in Primitive Roots g*(p)≪p 5/loglogp (1.4) uniformly for all large primes p⩾ 2. 1.3. Subsets of Primes with a Fixed Primitive Roots The main topic in Chapter 12 deals with an effective lower bound {p⩽x : ord(g)=p-1}≫x(logx) -1 (1.5) for the number of primes p⩽x with a fixed primitive root g≠±1,b 2 for all large number x⩾ 1. The current results in the literature have the lower bound {p⩽x : ord(g)=p-1}≫x(logx) -2, (1.6) and have restrictions on the minimal number of fixed integers to three or more, refer to Chapter 11, [HB86], and [MR88] for other details. The result is stated in Theorem 12.1. 1.4. Subsets of Composites with a Fixed Primitive Roots The main topic in Chapter 13 deals with an effective lower bound {N⩽x : ord(u) =λ(N)}≫x(log loglogx) -1 (1.7) for the number of composite N⩽x with a fixed primitive root u≠±1,v 2, and gcd(u,N)= 1, for all large number x⩾ 1.
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